{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XWOOPZ2XZDNFC3QEMS42SJA5H5","short_pith_number":"pith:XWOOPZ2X","canonical_record":{"source":{"id":"1603.06740","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-03-22T11:28:20Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"e4e67653242cba38988ea9c061608f229c365599c7ba08485181bf3ea156fdcd","abstract_canon_sha256":"518051f84ca1f953d0056a38e548ef833a6190b90caaf373147005b0f2522ec7"},"schema_version":"1.0"},"canonical_sha256":"bd9ce7e757c8da516e0464b9a9241d3f535eff315279ac50bce5a75cf4232c5f","source":{"kind":"arxiv","id":"1603.06740","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06740","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06740v1","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06740","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"XWOOPZ2XZDNF","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XWOOPZ2XZDNFC3QE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XWOOPZ2X","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XWOOPZ2XZDNFC3QEMS42SJA5H5","target":"record","payload":{"canonical_record":{"source":{"id":"1603.06740","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-03-22T11:28:20Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"e4e67653242cba38988ea9c061608f229c365599c7ba08485181bf3ea156fdcd","abstract_canon_sha256":"518051f84ca1f953d0056a38e548ef833a6190b90caaf373147005b0f2522ec7"},"schema_version":"1.0"},"canonical_sha256":"bd9ce7e757c8da516e0464b9a9241d3f535eff315279ac50bce5a75cf4232c5f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:34.808191Z","signature_b64":"ZsfIzfyAqwmf8UumjG4fA3h/lTKcdmP6D9/50UE6mWN3FFwGhwOIiRmu85BrWuzOiZhb2PyreARc1WQmH5KqAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd9ce7e757c8da516e0464b9a9241d3f535eff315279ac50bce5a75cf4232c5f","last_reissued_at":"2026-05-18T01:18:34.807616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:34.807616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.06740","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9iPZcbwO0T6alb1wg4k0IW0MLKnnVPSvnrj3Zg6mxuhmc2oedkFsoahqcx8pmT6d+YNH/z7vc8zrO9y/wAwKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:07:39.198714Z"},"content_sha256":"8c45d90b442b0035cc74a6caf82fbe0c0e28d267b3aac75677b863a81d2a5e65","schema_version":"1.0","event_id":"sha256:8c45d90b442b0035cc74a6caf82fbe0c0e28d267b3aac75677b863a81d2a5e65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XWOOPZ2XZDNFC3QEMS42SJA5H5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Grothendieck's Riemann-Roch Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.KT","authors_text":"Alberto Navarro","submitted_at":"2016-03-22T11:28:20Z","abstract_excerpt":"We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\\otimes \\bar L)=c_1(L)+c_1(\\bar L)-c_1(L)c_1(\\bar L)$. Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded $K$-theory $GK^\\bullet (X)\\otimes \\mathbb{Q}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06740","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:18:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NJzPmNrfWjqMqORNMqajF0M2JVPDjfSYp1zrozkPojOu1MT22KwblCb5D60LaGSPhJmn7HQONMCcmBELV9VOCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:07:39.199340Z"},"content_sha256":"dc97639ddc7ac157cf7bc98d0a988d4a52b92ecbe68464f25fccb1d39881e3d1","schema_version":"1.0","event_id":"sha256:dc97639ddc7ac157cf7bc98d0a988d4a52b92ecbe68464f25fccb1d39881e3d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/bundle.json","state_url":"https://pith.science/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T00:07:39Z","links":{"resolver":"https://pith.science/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5","bundle":"https://pith.science/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/bundle.json","state":"https://pith.science/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XWOOPZ2XZDNFC3QEMS42SJA5H5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XWOOPZ2XZDNFC3QEMS42SJA5H5","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"518051f84ca1f953d0056a38e548ef833a6190b90caaf373147005b0f2522ec7","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-03-22T11:28:20Z","title_canon_sha256":"e4e67653242cba38988ea9c061608f229c365599c7ba08485181bf3ea156fdcd"},"schema_version":"1.0","source":{"id":"1603.06740","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06740","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06740v1","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06740","created_at":"2026-05-18T01:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"XWOOPZ2XZDNF","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XWOOPZ2XZDNFC3QE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XWOOPZ2X","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:dc97639ddc7ac157cf7bc98d0a988d4a52b92ecbe68464f25fccb1d39881e3d1","target":"graph","created_at":"2026-05-18T01:18:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\\otimes \\bar L)=c_1(L)+c_1(\\bar L)-c_1(L)c_1(\\bar L)$. Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded $K$-theory $GK^\\bullet (X)\\otimes \\mathbb{Q}$.","authors_text":"Alberto Navarro","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-03-22T11:28:20Z","title":"On Grothendieck's Riemann-Roch Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06740","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8c45d90b442b0035cc74a6caf82fbe0c0e28d267b3aac75677b863a81d2a5e65","target":"record","created_at":"2026-05-18T01:18:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"518051f84ca1f953d0056a38e548ef833a6190b90caaf373147005b0f2522ec7","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-03-22T11:28:20Z","title_canon_sha256":"e4e67653242cba38988ea9c061608f229c365599c7ba08485181bf3ea156fdcd"},"schema_version":"1.0","source":{"id":"1603.06740","kind":"arxiv","version":1}},"canonical_sha256":"bd9ce7e757c8da516e0464b9a9241d3f535eff315279ac50bce5a75cf4232c5f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd9ce7e757c8da516e0464b9a9241d3f535eff315279ac50bce5a75cf4232c5f","first_computed_at":"2026-05-18T01:18:34.807616Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:34.807616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZsfIzfyAqwmf8UumjG4fA3h/lTKcdmP6D9/50UE6mWN3FFwGhwOIiRmu85BrWuzOiZhb2PyreARc1WQmH5KqAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:34.808191Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06740","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c45d90b442b0035cc74a6caf82fbe0c0e28d267b3aac75677b863a81d2a5e65","sha256:dc97639ddc7ac157cf7bc98d0a988d4a52b92ecbe68464f25fccb1d39881e3d1"],"state_sha256":"b9b608d66d4e8efc70d4c83e78ed89c579be76475e9413d368e4feb73d3dc145"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P+XeMmYHC3HALB3UJShgoHF3QUV69xBMEShUkCPJI5BW2TB1Sq8I0eWE3Pl9gHdldMSlZHrbGI+a5GfvLRnjDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:07:39.202781Z","bundle_sha256":"28511ca6ccd9fda96a1bc846456efec8d113ae94f9d008b700d761aa10f692d8"}}